Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Average Error: 6.3 → 2.6

Time: 3.7s

Precision: binary64

Math

\[x + \frac{y \cdot \left(z - x\right)}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.69623630598695 \cdot 10^{-44} \lor \neg \left(t \le 1.8302059170575166 \cdot 10^{-153}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y \cdot \left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right)}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\right) \cdot \frac{1}{\sqrt[3]{t}}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (y * ((double) (z - x)))) / t))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((t <= -5.69623630598695e-44) || !(t <= 1.8302059170575166e-153))) {
		VAR = ((double) (x + ((double) (y * ((double) (((double) (z - x)) / t))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) (y * ((double) (((double) cbrt(((double) (z - x)))) * ((double) cbrt(((double) (z - x)))))))) / ((double) cbrt(t)))) * ((double) (((double) cbrt(((double) (z - x)))) / ((double) cbrt(t)))))) * ((double) (1.0 / ((double) cbrt(t))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	2.1
Herbie	2.6

\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

1. Split input into 2 regimes

2. if t < -5.69623630598695e-44 or 1.8302059170575166e-153 < t

   1. Initial program 7.5

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 7.5

      \[\leadsto x + \frac{y \cdot \left(z - x\right)}{\color{blue}{1 \cdot t}}\]

   4. Applied times-frac 2.6

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - x}{t}}\]

   5. Simplified 2.6

      \[\leadsto x + \color{blue}{y} \cdot \frac{z - x}{t}\]

   if -5.69623630598695e-44 < t < 1.8302059170575166e-153

   1. Initial program 2.4

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 3.4

      \[\leadsto x + \frac{y \cdot \left(z - x\right)}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

   4. Applied times-frac 8.7

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{t}}}\]
   5. Using strategy rm

   6. Applied div-inv 8.7

      \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\left(z - x\right) \cdot \frac{1}{\sqrt[3]{t}}\right)}\]

   7. Applied associate-*r* 3.7

      \[\leadsto x + \color{blue}{\left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(z - x\right)\right) \cdot \frac{1}{\sqrt[3]{t}}}\]

   8. Simplified 7.4

      \[\leadsto x + \color{blue}{\left(\frac{y}{\sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{t}}\right)} \cdot \frac{1}{\sqrt[3]{t}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 7.4

       \[\leadsto x + \left(\frac{y}{\sqrt[3]{t}} \cdot \frac{z - x}{\color{blue}{1 \cdot \sqrt[3]{t}}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

   11. Applied add-cube-cbrt 7.6

       \[\leadsto x + \left(\frac{y}{\sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}}}{1 \cdot \sqrt[3]{t}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

   12. Applied times-frac 7.6

       \[\leadsto x + \left(\frac{y}{\sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{1} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\right)}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

   13. Applied associate-*r* 2.6

       \[\leadsto x + \color{blue}{\left(\left(\frac{y}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{1}\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\right)} \cdot \frac{1}{\sqrt[3]{t}}\]

   14. Simplified 2.5

       \[\leadsto x + \left(\color{blue}{\frac{y \cdot \left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right)}{\sqrt[3]{t}}} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.69623630598695 \cdot 10^{-44} \lor \neg \left(t \le 1.8302059170575166 \cdot 10^{-153}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y \cdot \left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right)}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\right) \cdot \frac{1}{\sqrt[3]{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (neg z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))